A256892 - OEIS

%I #18 Jun 28 2018 08:42:34

%S 1,0,1,0,3,1,0,13,9,1,0,73,79,18,1,0,501,755,265,30,1,0,4051,7981,

%T 3840,665,45,1,0,37633,93135,57631,13580,1400,63,1,0,394353,1192591,

%U 911582,274141,38290,2618,84,1,0,4596553,16645431,15285313,5633922,999831,92358,4494,108,1

%N Triangular array read by rows, the matrix product of the unsigned Lah numbers and the Stirling set numbers, T(n,k) for n>=0 and 0<=k<=n.

%C Also the Bell transform of A000262(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 29 2016

%F T(n+1,1) = A000262(n).

%F T(n+1,n) = A045943(n).

%F Row sums are A084357.

%e Triangle starts:

%e 1;

%e 0,    1;

%e 0,    3,    1;

%e 0,   13,    9,    1;

%e 0,   73,   79,   18,   1;

%e 0,  501,  755,  265,  30,  1;

%e 0, 4051, 7981, 3840, 665, 45, 1;

%p # The function BellMatrix is defined in A264428.

%p BellMatrix(n -> simplify(hypergeom([-n, -n-1], [], 1)), 9); # _Peter Luschny_, Jan 29 2016

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[Function[n, HypergeometricPFQ[{-n, -n-1}, {}, 1]], rows = 12];

%t Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%o def Lah(n, k):

%o     if n == k: return 1

%o     if k<0 or  k>n: return 0

%o     return (k*n*gamma(n)^2)/(gamma(k+1)^2*gamma(n-k+1))

%o matrix(ZZ, 8, Lah) * matrix(ZZ, 8, stirling_number2)  # as a square matrix

%Y See also A088814 and A088729 for variants based on an (1,1)-offset of the number triangles. See A131222 for the product Lah * Stirling-cycle.

%Y A079640 is an unsigned matrix inverse reduced to an (1,1)-offset.

%Y Cf. A000262, A045943, A084357, A088729, A088814.

%K nonn,tabl,easy

%O 0,5

%A _Peter Luschny_, Apr 12 2015